I don't think you play him because he's the scholarship goalie. I think you play him because he's, at least theoretically, the most talented goalie. You brought him in to be the starter, he's been terrible, but he's only played back-to-back games what? One time this year? Maybe he needs to get in a groove.
Do you give a guy more playing time if he hasn't earned it? Not typically, but it's not like the other guys have earned the job either. You'd have problems in the room if one of the goalies was playing well and they put in Rutledge because he was the scholarship guy, but I don't think anyone could really fault them for saying "Okay, this season has gone terribly. Rutledge was going to be the starter. It didn't work out in the early going, but nothing we've tried has worked. We're going to give him another chance and see if he can play like the guy we thought we were getting."
You have a point about "Giving up 5 goals a game isn't going to help him", and that's where they have to weigh the kid's psyche and use some information about him that the coaches should be aware of that the public doesn't have. I could see shutting him down for the year and rebooting next year with hopefully a clean slate, but I could also see playing him if you think he can give you anything.
I'm a fan of experimenting over the next 10-12 games so that by the time the tournament rolls around, hopefully they're playing their best hockey. They have not killed the tournament streak yet. Even if they lose the next 12 they're going to have a shot to win their way in. The goal should be to do whatever they have to do to be playing their best hockey in 6 weeks' time.
Rutledge has been awful, but he's also got a track record of success against quality competition. If there's one guy out of the three that's capable of getting hot, it's probably him, don't you think?